Description
1. (25 pts) Consider the following problem which was discussed in class:
Suppose that we have a patient who was just tested for a particular disease and
the test came out positive. We know that one in every thousand people has this
disease. We also know that the test is not reliable: it has a false positive rate of 3%
and a false negative rate of 6%. Our goal is then to assess our belief in the patient
having the disease given that the test came out positive. If we let the propositional
variable D stand for the patient has the disease, and the propositional variable T
stand for the test came out positive, our goal is then to compute P r(D|T).
We considered a version of this problem in class. You may also recall being surprised that
P r(D|T) was somewhat lower than expected. The goal of this question is then to identify
conditions under which this probability will be no less than .30. You will need to find the
answer to this by constructing a Bayesian Network and using the sensitivity analysis engine
of SamIam. You need to turn in:
• Your complete Bayesian network (structure and CPTs) in test.net file.
• A constraint on each of the following, which is sufficient to ensure that P r(D|T) ≥ 0.3:
The prior probability of having the disease, the false positive for the test, and the false
negative for the test.
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2. (25 pts) Figure 1 shows a Bayesian network classifier that predicts if a movie will be a boxoffice success. It takes in 4 binary inputs: S (Original Screenplay), G (Great Cinematography),
F (Famous Cast), and M (Marketing), and outputs a binary classification: C = 0 (failure)
or C = 1 (success). For convenience, the parameters of the Bayesian network are not shown
but the decision function of the classifier is given in Table 1. Construct the reduced decision
graph for this decision function, using the variable order: S, G, F, M.
C
S
N1
G F M
N2
Figure 1: A Bayesian network classifier for movie success.
S G F M C
1 – – – – –
2 – – – + –
3 – – + – +
4 – – + + +
5 – + – – –
6 – + – + –
7 – + + – +
8 – + + + +
9 + – – – –
10 + – – + –
11 + – + – –
12 + – + + +
13 + + – – –
14 + + – + –
15 + + + – +
16 + + + + +
Table 1: The decision function computed by the classifier in Figure 1.
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3. (25 pts) Using the decision graph in Figure 2, answer the following explanation queries.
(a) Consider a movie that is an original screenplay and has great cinematography, a famous
cast, and poor marketing {S = 1, G = 1, F = 1, M = 0}. Identify a largest set of
features that can be turned off (0) without changing the decision on this instance.
(b) Consider a movie that is an original screenplay and has poor cinematography, a famous
cast, and good marketing {S = 1, G = 0, F = 1, M = 1}. Identify a smallest set
of features α that renders the remaining features β irrelevant to the decision on this
instance. That is, if we fix features α to their current values, we can change the values
of features β arbitrarily without changing the current decision.
1 0
S
G G
F F
M
Figure 2: An ordered decision graph.
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4. (25 pts) Consider the Bayesian network in Figure 3 (supplied as a genetics.net).
Our physical traits, also called phenotypes are known to be caused by genetic
mutations such as single nucleotide polymorphisms (SNPs). It can be difficult to
identify if it’s a SNP that causes a phenotype or if it’s caused by something else
(which we call environmental effects). The following Bayesian network looks at four
different possible effects: 2 independent SNPs, sex, and being over 60. Each has a
probability of being present as well as a probability of causing the 2 phenotypes:
high BMI and high blood pressure where BMI can also effect Blood Pressure.
Use this network to answer the following queries:
(a) What are the prior marginal distributions for BMI and for blood pressure?
(b) What is the posterior marginal distribution for ‘over 60’ assuming either high BMI or
high blood pressure? (you will need to add an additional node to the network to answer
this query)
(c) If we know that SNP 2 is causal, what is the MAP for blood pressure and BMI?
(d) If we know that both SNPs are present and are causal, what is the MPE?
Figure 3: A Bayesian network.
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